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A364690
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Prime powers q such that there does not exist an elliptic curve E over GF(q) with cardinality q + 1 + floor(2*sqrt(q)).
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1
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128, 2048, 2187, 16807, 32768, 131072, 524288, 1953125, 2097152, 8388608, 14348907, 48828125, 134217728, 536870912, 30517578125, 549755813888, 847288609443, 2199023255552, 19073486328125, 140737488355328, 562949953421312, 36028797018963968, 144115188075855872, 450283905890997363
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OFFSET
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1,1
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COMMENTS
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By Hasse's theorem, every elliptic curve E over GF(q) has cardinality at most q + 1 + floor(2*sqrt(q)). Moreover, for every prime power q, there exists an elliptic curve E over GF(q) with cardinality at least q + floor(2*sqrt(q)). Thus these are the prime powers q for which A005523(n) = q + floor(2*sqrt(q)), where q = A246655(n).
By a theorem of Deuring and Waterhouse, these are exactly the prime powers q = p^k such that q is not prime, q is not a square, and p divides floor(2*sqrt(q)).
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LINKS
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EXAMPLE
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The first few values of the sequence (factorized) are 2^7, 2^11, 3^7, 7^5, 2^15, 2^17, 2^19, 5^9, 2^21, 2^23, 3^15, 5^11, 2^27, 2^29, ...
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PROG
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(Sage)
for q in range(1, 100000):
if Integer(q).is_prime_power():
p = Integer(q).prime_factors()[0]
if (floor(2*sqrt(q))%p == 0) and (not Integer(q).is_square()) and (q != p):
print(q)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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